Why did John Tyler not have a vice president?

Great question! We have grown accustomed to seeing the president with a vice president. That has not always been the case. There have been eighteen presidents that did not have a vice president.
In the early years of the republic, the role of vice president was not well defined—and was not as prestigious or considered a path to the White House—like it is today. Daniel Webster turned down the job in 1839. He was quoted as saying, "I do not propose to be buried until I am dead." Theodore Roosevelt once remarked, "I would a great deal rather be anything, say professor of history, than Vice President."
Tyler ascended to the presidency after the death of President William Henry Harrison, who died from pneumonia. Tyler has the distinction of being the first vice president to become president after the death of a president. Like many political contemporaries of his time, Tyler had a political nickname—"His Accidency" —for becoming president only through the death of Harrison. And, it is important to note, until the Twenty-Fifth Amendment to the Constitution was passed, it was not clear what should happen in if the president died or was unable to serve. During Tyler's time, being vice president did not entail the obligation of fulfilling the office of president in the event that the president could not carry out his duties, as it does in now.
There are two possible explanations, historians believe, as to why Tyler never chose a vice president. The first is—since the Constitution did not define the role of vice president or how to resolve the vacant office of the president—Tyler didn't see any reason to have one. The second reason is that Tyler was not elected to the office of president; he may have thought that he did not have the authority to choose someone to hold the office of vice president.
Given the number of presidents who did not have vice presidents (and the lack of respect the office commanded) another possible explanation is that people may not have been interested in the job.
https://forgottenhistoryblog.com/18-american-presidents-didnt-have-a-vice-president-for-all-or-some-of-their-terms/

https://www.history.com/topics/us-presidents/john-tyler

https://www.whitehouse.gov/about-the-white-house/presidents/john-tyler/